As human beings, are characters are pretty hard on themselves, blaming themselves for a lot of the things that happen to and around them. Once they realize the need for self-forgiveness, after a lifetime spent internalizing shame and guilt, it's often easier said than done. Individual circumstances will determine how easy or difficult this process is, including whether the character was truly to blame, the impact their actions had on others, if there are other long-lasting effects from their actions, if others also blame them, and a host of other factors. Take these into consideration when planning your character's journey to self-forgiveness.

Forgiving oneself for a past mistake
Choosing to let go of self-blame over one's part in an accident or tragedy
Forgiving oneself for holding back information that others deserved to know
Forgiving oneself for one's role in a feud that impacted others (children, a spouse, family members, etc.) which impacted their relationships, too
Choosing to let go of self-blame at responding with anger in the past, which led to missed opportunities and unhappiness for everyone involved
Forgiving oneself for keeping secrets from another to "protect them" even though it was wrong to do so
Forgiving oneself for past behavior that hurt other people

There are five basic human needs that, when missing from a character's life, could motivate them to pursue this goal. The following needs are all possibilities, but only one of them should be the primary driver for any given character. For more information on the relationship between human needs and outer motivations, please see this Character Motivation tutorial.

Esteem and Recognition: When a character's self-respect reaches a low or they realize that they're not respected as they should be by others, they may choose to do the hard work to forgive themselves.
Love and Belonging: Many times, the character's ideas about themselves are the very things that keep them from succeeding in relationships. If they lack love and belonging, they may turn to this goal as a way of filling that void.
Safety and Security: When an inability to forgive leads to self-hatred and -loathing, a character may become a danger to themselves, via destructive behaviors such as addictions, reckless behaviors, self-harm, and engaging in toxic relationships. If their safety and security become threatened, the character may choose to explore self-acceptance.